3.1824 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{(1-2 x)^{3/2}}{275 (5 x+3)}-\frac{3}{25} (1-2 x)^{3/2}+\frac{26}{275} \sqrt{1-2 x}-\frac{26 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

[Out]

(26*Sqrt[1 - 2*x])/275 - (3*(1 - 2*x)^(3/2))/25 - (1 - 2*x)^(3/2)/(275*(3 + 5*x)
) - (26*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi [A]  time = 0.0909177, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(1-2 x)^{3/2}}{275 (5 x+3)}-\frac{3}{25} (1-2 x)^{3/2}+\frac{26}{275} \sqrt{1-2 x}-\frac{26 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3 + 5*x)^2,x]

[Out]

(26*Sqrt[1 - 2*x])/275 - (3*(1 - 2*x)^(3/2))/25 - (1 - 2*x)^(3/2)/(275*(3 + 5*x)
) - (26*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi in Sympy [A]  time = 9.23169, size = 61, normalized size = 0.82 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{275 \left (5 x + 3\right )} + \frac{26 \sqrt{- 2 x + 1}}{275} - \frac{26 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**2*(1-2*x)**(1/2)/(3+5*x)**2,x)

[Out]

-3*(-2*x + 1)**(3/2)/25 - (-2*x + 1)**(3/2)/(275*(5*x + 3)) + 26*sqrt(-2*x + 1)/
275 - 26*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1375

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Mathematica [A]  time = 0.0884846, size = 58, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} \left (30 x^2+15 x-2\right )}{25 (5 x+3)}-\frac{26 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3 + 5*x)^2,x]

[Out]

(Sqrt[1 - 2*x]*(-2 + 15*x + 30*x^2))/(25*(3 + 5*x)) - (26*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(25*Sqrt[55])

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Maple [A]  time = 0.017, size = 54, normalized size = 0.7 \[ -{\frac{3}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{12}{125}\sqrt{1-2\,x}}+{\frac{2}{625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{26\,\sqrt{55}}{1375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^2,x)

[Out]

-3/25*(1-2*x)^(3/2)+12/125*(1-2*x)^(1/2)+2/625*(1-2*x)^(1/2)/(-6/5-2*x)-26/1375*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50162, size = 96, normalized size = 1.3 \[ -\frac{3}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{13}{1375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12}{125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="maxima")

[Out]

-3/25*(-2*x + 1)^(3/2) + 13/1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) + 12/125*sqrt(-2*x + 1) - 1/125*sqrt(-2*x + 1)/(5*x
+ 3)

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Fricas [A]  time = 0.218546, size = 93, normalized size = 1.26 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (30 \, x^{2} + 15 \, x - 2\right )} \sqrt{-2 \, x + 1} + 13 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{1375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="fricas")

[Out]

1/1375*sqrt(55)*(sqrt(55)*(30*x^2 + 15*x - 2)*sqrt(-2*x + 1) + 13*(5*x + 3)*log(
(sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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Sympy [A]  time = 49.6527, size = 187, normalized size = 2.53 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25} + \frac{12 \sqrt{- 2 x + 1}}{125} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{125} + \frac{128 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**2*(1-2*x)**(1/2)/(3+5*x)**2,x)

[Out]

-3*(-2*x + 1)**(3/2)/25 + 12*sqrt(-2*x + 1)/125 - 44*Piecewise((sqrt(55)*(-log(s
qrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*
(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605,
(x <= 1/2) & (x > -3/5)))/125 + 128*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2
*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/5
5, -2*x + 1 < 11/5))/125

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GIAC/XCAS [A]  time = 0.212767, size = 100, normalized size = 1.35 \[ -\frac{3}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{13}{1375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{12}{125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="giac")

[Out]

-3/25*(-2*x + 1)^(3/2) + 13/1375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 12/125*sqrt(-2*x + 1) - 1/125*sqrt(-2*x +
1)/(5*x + 3)